Fault Mechanics
❍ Laboratory
Pore pressure
scale
Lowers normal
stress, moves
stress circle to left
❍ Doesn’
Doesn’t change
shear
❍ Deviatoric stress
not affected
❍ This example:
failure will be by
tensile cracks
❍
❍Mohr’
Mohr’s circles in mohr detail
❍ Failure
at large scale
❍Anderson theory
❍Earthquakes without the shaking
❍ Brittle
deformation and crack theory
❍ Dislocations
❍ Next
week - rheology
Normal and shear stresses
❍ Vertical
σ1
❍ Horizontal
❍
❍
Ordinary space
σ3
Arbitrary angle
is θ
2D picture, but
σ2 = σ3
Laboratory scale
❍ Much
of this is done by rock mechanics
specialists
❍ Engineering geology needs their input
❍ We use bits of it
❍ Low and high pressure apparatus
Back to Mohr diagram for stress
❍ A whole
different
space - normal
versus shear
stresses
❍ This should be
very familiar
❍ Now, mapping
back onto σ1 - σ3
at some angle to σ1
identify that plane by
its pole (perpendicular)
❍ Maximum shear stress
across that plane is at 45º
❍ Normal stress still more
than half-way at 45º
❍ Plane
❍ We
space, we get ……
Mohr stress
Three dimensional Mohr
Ordinary
❍2
points same shear, different normal
points equal normal, opposite shear
❍ Complementary angles
❍2
For triaxial state
of stress, 3
nested circles
❍ This makes the
angles of
maximum
normal/shearme
ssy
❍ Best done with
tensors
❍
Mohr
Coulomb failure envelope
Shear stress
!
!c = !0 + " tan$
!0
$
2#
"2
P
"1
Normal stress
"
❍ Optimum combination
❍ “Coefficient of
of normal/shear
internal friction”
friction”
1
Coulomb failure envelope
❍ Angle
of failure
❍Tangent to failure envelope
❍Angle relative to pole (perpendicular) to σ1
Shear stress
!
!c = !0 + " tan$
!0
$
2#
"2
90 ! " = 180 ! 2# critical
90 + "
# critical =
2
"1
P
Normal stress
"
For ! = 30°, a typical value,
" critical = 60°
Mohr-Coulomb failure envelope
The envelope not
actually linear
Failure
mechanism
changes as
function of
pressure
❍ Still, more or less
what you should
by used to
❍
❍
❍A
series of mechanisms
pressure relative
to rock strength determines
which
❍ Confining
❍Tensile fracture
by σ1
plane parallel to σ1
❍ Dominated
❍ Fracture
❍Transitional
❍ It’
It’s
What do we mean
by failure?
❍ Failure
What do we mean
by failure?
cracking up
Cracks
What do we mean by failure?
II
❍Coulomb shear failure
❍ Note
the angle on the
Coulomb cylinder: be sure
you understand why it
behaves that way
❍Brittle-ductile transition
❍ Almost
always gradual
❍Plastic yielding
❍ Von
Mises criterion
What do we mean by failure?
Macroscopic fault mechanics:
At the outcrop to regional scale
Structural geologists
Seismologists
Seismologists
❍ The world is not simple…
simple…
❍ Anderson theory
❍
Shear zones
❍ Actual
geology more complicated than
laboratory, even though that is bad enough
❍
❍ Vertical stress is given by ρgh in each case
❍ More
❍ We
or less lithostatic
drop the lithostatic, only deal with deviatoric stress
❍ What kind of fault you get depends on the relative values
of the three deviatoric stresses
❍ Thrust, normal, strike slip
2
Map and cross section views
Anderson theory: simplified version
Anderson
theory
❍ Faults
are seldom
straight
❍ And the structures differ
with depth
In effect, style of fault slip is controlled by which
stress direction is least confined
Fault angle not specified by Anderson theory
❍ Room problems, failure law, fluids, heterogeneities
can all enter
❍
❍
Faults shouldn’
shouldn’t cross?
What does Anderson predict?
❍ Assume rocks
are riddled with faults
❍Slip at angle with minimum tectonic stress
❍ Not
the same as
Mohr-Coulomb
❍ Coefficient of
friction controls
fault angle
❍ Only
❍ You
a problem for brittle fracture
can also crush intersections
Friction on faults
Friction on faults
Coefficient of friction ! yx = fs! yy
Does Anderson theory work?
❍ If
you conflate Mohr-Coulomb theory at
tan φ ≈ 30º
30º with Anderson theory, you
predict normal faults dipping at 60º, thrust
faults at 30º, and strike-slip faults at 30º to
the regional stress field.
❍ But……
But……
❍Large thrust faults dip as little as 2º
❍San Andreas perpendicular to maximum
horizontal stress direction
❍Worst of all, large normal faults dip as little as 6º
Brittle deformation and cracks
❍ Dig
now into microscopic scale
- don’
don’t let the wings fall off
planes
❍ Modes of cracks
❍ Engineering
❍Tensile
❍2 shear modes
❍Stress intensity factors
❍ Wing
cracks
3
Wing cracks
Tensile and shear cracks
Mostly look at brittle deformation
Tensile - pull apart (Mode I)
Shear II - sliding parallel to crack (Mode II)
❍ Shear III - tearing perpendicular to crack (Mode III)
Stress concentration factors
❍
❍
❍
Down to even more
microscopic level:
Dislocations
Single defects can move
❍ Aids
❍ And
❍To some extent
can nucleate
line defects, etc.
Line defects (dislocations)
series of defects arranged in a
more or less linear array
❍ Two types
❍Edge dislocations
❍Screw dislocations
❍ Can
get serious deformation when these
move through lattice
diffusion
deformation
❍ Also
Explained (maybe)
Point and line crystal flaws
❍ A whole
Mode II and III cracks are unstable
❍ Tensile stress concentration bends them out of plane
❍ You can hear this happening
❍
Edge dislocations
Lattice plane stops
abruptly along line
❍ Line vector is tip of
dislocation
❍ Burgers vector: points
in direction of offset of
lattice plane
❍ Slip at right angles to
line vector, parallel to
Burgers vector
❍
Point defects
❍ Single
atoms out
of place
❍ Extras, deficits
❍ Can move through
lattice
Edge dislocations move
❍ Have
to break, reform bonds at dislocation
tip as planes rearrange
❍ Always an odd
plane out
4
Screw locations on the move
❍ Incremental
slippage
as dislocation moves
❍ Creates ramp in
lattice planes
❍ Dislocation glide
Both can coexist
❍ Edge
❍ Dislocations
quartz
The real world - not a pretty
picture
The world seen in a corn cob
on front face, screw on left
in
Rheology - a Continuum
Mechanics Viewpoint
❍ A mechanistic
view
❍Combine “springs”
springs” and “dashpots”
dashpots”
❍Represent elastic and viscous aspects of rock
❍Recoverable and permanent deformation
❍ Time
varying behavior
❍Temperature dependent
❍Can make quite complicated mixes
❍ More
of this next week
5